Should I apply to the School of Engineering and Applied Science (SEAS) or to the Department of Earth and Planetary Sciences (EPS)?

It makes no difference as far as the criteria for admission, the research you'll do, the stipend you'll get, the office you'll be in, the students you'll hang out with, or the general requirements for the Ph.D. The only differences are in (1) the courses you'll be asked to take (even there the difference is slight), (2) the teaching requirement (see question on teaching requirement below), (3) the administrators you'll interact with, and (4) the label of your Ph.D. SEAS students generally apply in the Environmental Science and Engineering (ESE) area, though I have had a few coming in through Applied Math or Applied Physics (the only difference here is in the courses you may be expected to take). The current graduate students in the group are split between SEAS and EPS. Browse through the SEAS and EPS web pages and decide where you'd feel most at home.